diff -r c563cf946497 -r 3a12e096f9a0 slides/slides02.tex
--- a/slides/slides02.tex	Sat Oct 03 15:21:06 2020 +0100
+++ b/slides/slides02.tex	Sun Oct 04 12:30:24 2020 +0100
@@ -344,7 +344,8 @@
 
 \end{frame}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-  
+
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{frame}[c]
@@ -1011,7 +1012,48 @@
 \end{frame}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[c]
+\frametitle{The Specification for Matching}
 
+\begin{bubble}[10cm]
+\large
+A regular expression \bl{$r$} matches a string~\bl{$s$} 
+provided:
+\begin{center}
+\bl{$s \in L(r)$} 
+\end{center}\medskip
+\end{bubble}
+
+
+\begin{center}
+\onslide<2>{\bl{\large$\forall\,r\,s.\;$}}
+\bl{$matcher\,s\,r$}  \;if and only if\  \bl{$s \in L(r)$} 
+\end{center}
+
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+  
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[c]
+\frametitle{nullable and \boldmath$der$}
+
+The central properties:\bigskip\bigskip
+
+\large
+\begin{quote}
+  \onslide<3>{\bl{$\forall r.\phantom{\,c}\;$}}%
+  \bl{$nullable(r)$} \;if and only if\; \bl{$[] \in L(r)$}
+\end{quote}\bigskip\bigskip\pause
+
+\begin{quote}
+  \onslide<3>{\bl{$\forall r\,c.\;$}}%
+  \bl{$L(der\,c\,r) \;=\; Der\,c\,(L(r))$}
+\end{quote}
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
   
   
 
@@ -1056,7 +1098,7 @@
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{frame}[c]
-\frametitle{Proofs about Rexp (3)}
+\frametitle{Proofs about Rexp}
 
 Assume \bl{$P(r)$} is the property:
 
@@ -1067,6 +1109,9 @@
 \end{frame}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
+\end{document}
+% below are slides for proving.
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{frame}[c]
 \frametitle{Proofs about Rexp (4)}